A variable line, drawn through the point of intersection of the straight lines x a + y b = 1 and x b + y a = 1 , meets the coordinate axes in A & B . Show that the locus of the mid point of A B is the curve 2 x y ( a + b ) = a b ( x + y ) .

The given equations of the lines are x a + y b = 1 and x b + y a = 1 ⇒ b x + a y - a b = 0 and a x + b y - a b = 0 . The equation of a line passing through the point of intersection of the lines L 1 = 0 and L 2 = 0 is L 1 + λ L 2 = 0 . Hence, the equation of a line passing through the point of intersection of the lines b x + a y - a b = 0 & a x + b y - a b = 0 is b x + a y - a b + λ a x + b y - a b = 0 ⇒ x b + λ a + y a + λ b - a b 1 + λ = 0 . To find the x -intercept of the line, put y = 0 , to get x b + λ a - a b 1 + λ = 0 ⇒ x = a b 1 + λ b + λ a . To find the y -intercept of the line, put x = 0 , to get y a + λ b - a b 1 + λ = 0 ⇒ y = a b 1 + λ a + λ b . Thus, the points on the axes are respectively A a b 1 + λ b + λ a , 0 and B 0 , a b 1 + λ a + λ b . The mid-point of the line segment joining the points x 1 , y 1 and x 2 , y 2 is x 1 + x 2 2 , y 1 + y 2 2 . Hence, the mid-point of the line segment joining A and B is h , k = a b 1 + λ 2 b + λ a , a b 1 + λ 2 a + λ b ⇒ h = a b 1 + λ 2 b + λ a , k = a b 1 + λ 2 a + λ b Now, to obtain the locus, we have to eliminate λ , From the above relation, we have 2 h b + λ a = a b 1 + λ and 2 k a + λ b = a b 1 + λ ⇒ 2 h b + 2 h λ a = a b + a b λ and 2 k a + 2 k λ b = a b + a b λ ⇒ λ 2 a h - a b = a b - 2 h b and λ 2 k b - a b = a b - 2 k a ⇒ λ = b a - 2 h a 2 h - b and λ = a b - 2 k b 2 k - a On comparing, we get b a - 2 h a 2 h - b = a b - 2 k b 2 k - a ⇒ b 2 a - 2 h 2 k - a = a 2 b - 2 k 2 h - b ⇒ 2 a b 2 k - a 2 b 2 - 4 b 2 k h + 2 a b 2 h = 2 a 2 b h - a 2 b 2 - 4 a 2 h k + 2 a 2 b k ⇒ 2 a b 2 k + 2 a b 2 h - 2 a 2 b h - 2 a 2 b k = 4 b 2 h k - 4 a 2 h k ⇒ 2 a b 2 k + h - 2 a 2 b h + k = 4 h k b 2 - a 2 ⇒ 2 a b k + h b - a = 4 h k b - a b + a ⇒ a b h + k = 2 h k a + b Now, to get the locus replace h , k by x , y , to get a b x + y = 2 x y a + b .